Question

If \(\alpha,\beta\) are the roots of the quadratic equation \[ x^2-3x+1=0, \] then the value of \(\alpha^3+\beta^3\) is:

• A. \(9\)
• B. \(18\)
• C. \(21\)
• D. \(27\)

Hint:

For roots \(\alpha,\beta\) of a quadratic equation, always remember: \[ \alpha+\beta=-\frac{b}{a}, \qquad \alpha\beta=\frac{c}{a}. \] Many questions involving powers of roots can be solved without finding the actual roots.

Answer 1

The Correct Option is B

Concept: Whenever a question involves roots of a quadratic equation and asks for expressions such as \(\alpha^2+\beta^2\), \(\alpha^3+\beta^3\), \(\frac{1}{\alpha}+\frac{1}{\beta}\), etc., it is generally unnecessary to find the roots explicitly. Instead, we use Vieta's formulas which directly relate the roots of a quadratic equation to its coefficients. For a quadratic equation \[ ax^2+bx+c=0, \] having roots \(\alpha\) and \(\beta\), \[ \alpha+\beta=-\frac{b}{a} \] and \[ \alpha\beta=\frac{c}{a}. \] After obtaining these values, standard algebraic identities can be used to evaluate higher powers of the roots. This approach is faster, more elegant, and avoids lengthy calculations.

Step 1: Identify the given quadratic equation. The given equation is \[ x^2-3x+1=0. \] Comparing with the standard form \[ ax^2+bx+c=0, \] we obtain \[ a=1,\qquad b=-3,\qquad c=1. \]

Step 2: Apply Vieta's formulas. Using \[ \alpha+\beta=-\frac{b}{a}, \] we get \[ \alpha+\beta=-\frac{-3}{1}=3. \] Similarly, \[ \alpha\beta=\frac{c}{a} =\frac{1}{1} =1. \] Thus, \[ \alpha+\beta=3 \] and \[ \alpha\beta=1. \]

Step 3: Recall the identity for cubes. The standard identity is \[ \alpha^3+\beta^3 = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta). \] This identity allows us to calculate the required value directly.

Step 4: Substitute the known values. Substituting \[ \alpha+\beta=3 \] and \[ \alpha\beta=1, \] we obtain \[ \alpha^3+\beta^3 = 3^3 - 3(1)(3). \] \[ = 27-9. \] \[ = 18. \]

Step 5: Verification. The computed value is \[ 18. \] Since all calculations have been performed using exact identities and Vieta's formulas, the result is completely consistent with the given quadratic equation.

Step 6: Final Conclusion. Therefore, \[ \boxed{\alpha^3+\beta^3=18} \] Hence the correct answer is \[ \boxed{\text{Option (B)}}. \]